Preheat oven to 375 degrees F. Combine squash, stock, and milk in a medium saucepan; bring to a boil over medium-high heat. Reduce heat to medium; simmer until squash is tender when pierced with a fork, about 20 minutes. Remove from heat. Mash contents of saucepan; stir in nutmeg, cayenne, and salt, and season with black pepper. Stir to combine.

Meanwhile, bring a large pot of water to a boil. Add noodles; cook until al dente according to package instructions, about 8 minutes. Drain, and transfer to a large bowl; stir in squash mixture, cheddar, ricotta, and 2 tablespoons Parmesan.